MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isnzr Structured version   Unicode version

Theorem isnzr 18034
Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
isnzr.o  |-  .1.  =  ( 1r `  R )
isnzr.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
isnzr  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)

Proof of Theorem isnzr
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . 4  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
2 isnzr.o . . . 4  |-  .1.  =  ( 1r `  R )
31, 2syl6eqr 2516 . . 3  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
4 fveq2 5872 . . . 4  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
5 isnzr.z . . . 4  |-  .0.  =  ( 0g `  R )
64, 5syl6eqr 2516 . . 3  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
73, 6neeq12d 2736 . 2  |-  ( r  =  R  ->  (
( 1r `  r
)  =/=  ( 0g
`  r )  <->  .1.  =/=  .0.  ) )
8 df-nzr 18033 . 2  |- NzRing  =  {
r  e.  Ring  |  ( 1r `  r )  =/=  ( 0g `  r ) }
97, 8elrab2 3259 1  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652   ` cfv 5594   0gc0g 14857   1rcur 17280   Ringcrg 17325  NzRingcnzr 18032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-iota 5557  df-fv 5602  df-nzr 18033
This theorem is referenced by:  nzrnz  18035  nzrring  18036  drngnzr  18037  isnzr2  18038  isnzr2hash  18039  ringelnzr  18041  subrgnzr  18043  zringnzr  18627  chrnzr  18694  nrginvrcn  21326  ply1nzb  22649  zrhnm  28111  isdomn3  31368
  Copyright terms: Public domain W3C validator